Hi,
Since you wanted a simple answer, I will include a summary for you. If
you want further in-depth info, you can visit the web sites I compiled
for you below.
Summary:
An integer greater than one is called a "prime number" if its only
positive divisors are itself and one. The real importance of prime
numbers was understood with the discovery of the data encryption
methods used today. These encryption methods (for example, used to
encrypt secure Internet traffic, credit card info, etc.) stands on the
fact that any integer greater than one can be written as a product of
prime numbers.
A Mersenne Prime is a special case of a prime number. By definition,
if a prime number can be written as
2 ^ n - 1
[Note: 2^n notation means 2 to the power of n]
then it is said to be a "Mersenne Prime". The theorem that immediately
follows is that n is a prime number, too (proof below).
After this quick summary, you may want to visit web sites that deal
with the topic in more depth. You can find definition and importance
of prime numbers in the following pages (I included an abstract for
each site for convenience, but I advise you to go to the sites for
comprehensive explanations):
Sweepstakes: give the old lady a new look
http://www.carpiohelpdesk.com/CRM%20Articles/Sweepstakes_%20give%20the%20old%20lady/body_sweepstakes_%20give%20the%20old%20lady.htm
Prime numbers play an extremely important role in mathematics and are
used in numerous calculations (most known are factoring, greatest
common divisor, linear equation solving, etc.). But perhaps the most
important quality of prime numbers is the simplest one: any number
greater than one may be written as a product of prime numbers.
But their real importance for the computer world became evident around
1977, when R.L. Rivest, A. Shamir, and L. Adleman discovered a way to
encode messages in such a way that the code would be almost impossible
to break even if the method of encoding was public, i.e. known to
everybody.
The Prime Pages
http://www.utm.edu/research/primes/
An integer greater than one is prime if its only positive divisors are
itself and one.
Prime Number - from Mathworld
http://mathworld.wolfram.com/PrimeNumber.html
Because of their importance in encryption algorithms such as RSA
encryption, prime numbers can be important commercial commodities.
Why study Prime and Composite Numbers?
http://mathforum.org/library/drmath/view/57182.html
Every time you send a credit card number over the Internet, it gets
encrypted by your browser, and the encryption algorithm is based on
the theory of prime numbers.
The Mersenne Prime information can be found at the following web
sites:
Mersenne Primes: History, Theorems and Lists
http://www.utm.edu/research/primes/mersenne/index.html
Definition: When 2^n-1 is prime it is said to be a Mersenne prime.
[Note: 2^n notation means 2 to the power of n]
Proof of the theorem: "If 2^n-1 is prime, then so is n"
http://www.utm.edu/research/primes/notes/proofs/Theorem2.html
My search stategy to find the web sites:
"importance of prime numbers"
encryption "prime number"
"mersenne prime"
Hope this helps
Regards
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